Interface scaling in a two-dimensional porous medium under

PHYSICAL REVIEW E 66, 051603 共2002兲

Interface scaling in a two-dimensional porous medium under combined viscous, gravity, and capillary effects Yves Me´heust,1,2 Grunde Lo” voll,2,1 Knut Jo” rgen Ma˚lo” y,2 and Jean Schmittbuhl1 Laboratoire de Ge´ologie, E´cole Normale Supe´rieure, Paris, France 2 Department of Physics, University of Oslo, Oslo, Norway 共Received 6 July 2001; published 11 November 2002兲

1

We have investigated experimentally the competition between viscous, capillary, and gravity forces during drainage in a two-dimensional synthetic porous medium. The displacement of a mixture of glycerol and water by air at constant withdrawal rate has been studied. The setup can be tilted to tune gravity, and pressure is recorded at the outlet of the model. Viscous forces tend to destabilize the displacement front into narrow fingers against the stabilizing effect of gravity. Subsequently, a viscous instability is observed for sufficiently large withdrawal speeds or sufficiently low gravity components on the model. We predict the scaling of the front width for stable situations and characterize it experimentally through analyses of the invasion front geometry and pressure recordings. The front width under stable displacement and the threshold for the instability are shown, both experimentally and theoretically, to be controlled by a dimensionless number F which is defined as the ratio of the effective fluid pressure drop 共i.e., average hydrostatic pressure drop minus viscous pressure drop兲 at pore scale to the width of the fluctuations in the threshold capillary pressures. DOI: 10.1103/PhysRevE.66.051603

PACS number共s兲: 47.55.Mh, 47.20.Gv, 68.05.Cf, 47.53.⫹n

I. INTRODUCTION

Multiphase fluid flows in porous media have many important applications in geological engineering, including ground water flow modeling and oil recovery, where an increase in the recovery rate is obtained by injection of another fluid phase 关1– 8兴. In immiscible two phase flows, strong variations exist in the obtained displacement structures and dynamics, depending on the flow rates, wetting properties, viscosity ratios, and density differences of the involved fluids 关9,10兴. If one considers situations in which a nonwetting fluid invades a porous medium saturated with a wetting fluid 共drainage兲 with a higher viscosity, in the absence of gravity, two regimes can be distinguished. At very low flow rates, the viscous pressure drop across the porous medium is negligible in comparison to the inhomogeneity in the threshold capillary pressures inside the medium. The topology of the random porous medium causes the fluctuations in the pressure field needed for the nonwetting phase to invade new pores. The resulting displacement structure is controlled by capillary effects, and is well described by the invasion percolation algorithm. This capillary fingering regime has been studied extensively 关11–13兴. At high flow rates, viscous forces dominate capillary and gravitational effects. The displacement is stable or unstable depending on which phase is the most viscous 关4,14 –16兴. If the defending phase is the most viscous one, the displacement is unstable 关4,14,17,18兴. This viscous fingering regime has strong analogies to diffusion-limited aggregation 共DLA兲 patterns 关19–23兴. In the opposite case in which the invading fluid is the most viscous one, viscous effects stabilize the front 关16,24兴. However, most real reservoir systems are not flat and horizontal, and are therefore sensitive to gravity effects. If the two fluids involved have different densities, gravity forces modify the displacement structure dramatically 关25– 1063-651X/2002/66共5兲/051603共12兲/$20.00

28. Gravity causes hydrostatic pressure gradients that might stabilize or destabilize fluid motion. In the capillary fingering regime, stabilization of the fluid interface by gravity has been studied both theoretically and experimentally in two and three dimensional media 关25,26,29,30兴. Unstable buoyancy driven migration of a lighter fluid into a denser one has also been studied by use of experiments, computer simulations and theoretically 关27,28,31–33兴. All these studies of gravitational effects concentrate on situations where the flow rates are so low that capillary forces are large compared to viscous forces at pore scale. For such systems the obtained flow patterns are understood in terms of the competition between gravity forces and local nonhomogeneous capillary forces 关25,26兴. It should be noted that though the obtained displacement patterns could be modeled without considering viscosity, viscous effects have to be taken into account to understand the local dynamics during invasion 关34,35兴. The competition between gravity and viscous forces has been studied for immiscible flows in a Hele-Shaw cell without any porous medium. Saffman and Taylor have studied small perturbations of the displacement front, deriving a criterion for the instability 关36兴; but it is not obvious that the arguments developed for this particular geometry apply to flows in porous media, where the complexity of the flow boundary conditions plays a crucial role 关17兴. The deformation and breakup of nonwetting clusters by a viscous pressure field has also been studied for a system without gravity effects 关37兴. Situations where all three forces are significant have been studied for imbibition in a sand pack 关38兴, with a focus on the saturation of the nonwetting fluid, and for simulations of viscous fingering in a gravity field, at reservoir scale 关7,39兴. The purpose of this work is to address the scaling properties of an interface during a two phase flow in a porous medium when capillary, gravity and viscous forces play a comparable role. We have studied specifically the stabiliza-

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©2002 The American Physical Society

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FIG. 1. Sketch of the two-dimensional porous medium used in the experiments. The beads are 1 mm in diameter and the distance between the channels is L⫽350 mm. The model is sealed with silicone glue along the edges. The size of the model is L⫻L.

tion and destabilization of the displacement front obtained by injection of air from above in a two dimensional porous medium saturated with a wetting fluid. This system was studied systematically by tuning gravity and viscous forces: the gravitational component on the model could be varied by tilting the model at different angles, while the magnitude of viscous forces could be changed by varying the injection rate, which was typically high enough for viscous fingering to occur at low inclination angles. The present article is organized as follows. The experimental setup and the methods used for treating the data are described and discussed in Sec. II. In Sec. III, we discuss the competition between gravity forces, viscous forces and capillary forces from a theoretical point of view. The experimental results are presented in Sec. IV, which is followed by a discussion of the results in Sec. IV E and some concluding remarks in Sec. V. II. EXPERIMENTAL METHOD A. Experimental setup and procedure

The model used in the study is similar to the one used in 关14,15兴 and 关37兴. A two dimensional porous media is made by pouring glass beads of diameter a⫽1 mm on the sticky side of a 500⫻500 mm2 contact paper until no more beads stick to the surface. The excess beads are then removed so that the resulting porous matrix is a random monolayer of

beads. The contact paper is glued to a Plexiglas plate with milled in and outlet channels. The channels are parallel and separated by 350 mm. They are 5 mm wide, 8 mm deep and 350 mm long, which makes the size of the porous medium L⫻L, with L⫽350 mm. When the model is installed, the outlet channel is connected to a pump and a Honeywell 26PCA Flow-Through pressure sensor. The channels are cut open and the edges of the model are sealed and constrained with silicone glue. Then another sheet of contact paper is glued on top of the beads so as to make a ‘‘sandwich’’ 共see Fig. 1兲. The model is then placed 共beads down兲 on top of another Plexiglas plate with a ‘‘pressure cushion’’ to ensure that the model is kept in position and that it is always only one bead diameter thick. The porosity ␾ of the model is measured to be 0.63 and the permeability ␬ is measured to be ␬ ⫽0.0189⫻10⫺3 cm2 ⫽1948 darcy. The defending wetting fluid used in all our experiments is a 90–10 % by weight glycerol-water solution dyed with 0.1% Negrosine. Air is used as the invading nonwetting fluid. The wetting glycerol-water solution has a viscosity of ␮ w ⬇0.2 Pa s and a density of ␳ w ⫽1235 kg m⫺3 at room temperature. The corresponding parameters for the nonwetting air are ␮ nw ⫽1.9⫻10⫺5 Pa s and ␳ nw ⫽116 kg m⫺3 . The surface tension between these two liquids ␴ is 6.4⫻10⫺2 N m⫺1 . The temperature in the defending fluid is measured at the outlet of the model during each experiment to be able to estimate the viscosity. A ⫾1 °C uncertainty between this measured value and the temperature of the fluid inside the model is assumed, resulting in a ⫾2⫻10⫺2 Pa s uncertainty in the estimated viscosity. To get stable flow out of the model we use gravity to pump the defending fluid out of the model. This is done by leading the fluid out of the model and down to a reservoir on the floor below (⬃4 m height difference兲. The flow rate is controlled by a valve and measured during experiments with an electronic scale. Flow rate variations during an experiment are always smaller than 3% of Q. The Honeywell 26PCA Flow-Through pressure sensor measures the pressure in the outlet channel during experiments. By using the pressure in the outlet channel at breakthrough as a reference, we measure the pressure difference between the invasion front and the outlet channel during the experiment.

FIG. 2. Sketch of the experimental setup. The gravity component on the porous medium is varied by changing the tilt angle of the setup.

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INTERFACE SCALING IN A TWO-DIMENSIONAL . . .

FIG. 3. 共a兲 Comparison between the original picture of the flow structure and the extracted invasion front. The whole model (35 ⫻35 cm2 ) is represented. The withdrawal rate is 1.5 ml min⫺1 . The inclination angle is 20.3° 共experiment 38兲. This situation corresponds to an intermediate regime in which a significant viscosity-related fingering exists at some places in the displacement structure, while other parts of it exhibit a dynamics closer to capillary fingering. Anisotropic shapes of the interface and the trapped clusters are the signature of this crossover regime. 共b兲 Corresponding extracted black and white picture. The displacement structure appears white. Its aspect is related to the physical phenomena occurring during drainage. 共c兲 Black and white picture after removal of the trapped clusters. The front plotted in 共a兲 on top of the raw image has been obtained as the line separating the white from the black area in 共c兲.

The model is clamped to a light box that can be tilted to various inclination angles to vary the component of gravity on the porous medium. The component of gravity along the model g is given by g⫽g 0 sin(␣), where g 0 is the acceleration of gravity and ␣ is the inclination angle 共see Fig. 2兲. The pressure drop between the outlet of the model and the reservoir is much larger than that between the inlet and the outlet of the model. The flow rate is therefore controlled by the former pressure drop, and its dependence on the tilting of the experimental setup or on the volume of wetting fluid inside the model is very weak. Thus, gravity can be considered to be tuned independently of the flow rate. The displacement structure is visualized by illuminating the model with the light box and taking pictures with an attached Kodak DCS 420 CCD camera, which is controlled by a computer over a SCSI connection. Pictures are taken at regular time intervals during fluid displacement. They have a resolution of 1536⫻1024 pixels, with 8 bits gray levels, which provides a spatial resolution of 0.37 mm per pixel or ⬃2.56 pixels per pore. B. Image treatment 1. Front extraction

The raw images display a variety of gray scales. A region with very light gray shadings corresponding to air and others with dark shadings corresponding to dyed glycerol 关see Fig. 3共a兲兴 can be distinguished. Glass beads appear as medium gray spots randomly distributed in the picture. In order to extract the part of the model that is filled with air, all gray levels below a given threshold are set to black, while the others are set to white. The threshold chosen varies along the direction normal to the direction of drainage, to account for the enlightenment nonuniformity. During this thresholding process, glass beads are arbitrarily determined to be part of the white or of the black region. This does not affect the accuracy of the following analyses, since only scales larger than the typical pore size, a, are considered as significant. Geometrical information about the drainage process contained in the raw picture is passed to the resulting black and white picture 关see Fig. 3共b兲兴. We call the white region in this

picture displacement structure. Apart from included glass beads, it corresponds to the part of the model filled with air. The black islands inside the white area in the black and white picture are clusters of wetting fluid that have been trapped in air during drainage, as two neighboring fingers of the fronts advancing ahead of the rest of the front have merged. Removal of the trapped clusters is achieved by painting white all black islands in the picture except the largest one, leaving only one continuous white area and one continuous black area 关see Fig. 3共c兲兴. The latter is then separated from the former by a continuous line, the displacement front, which is the limit for the penetration of air inside glycerol. In what follows, we often simply call it the front. In Fig. 3共a兲, the front is shown in black on top of the raw image. Visual inspection shows a very good consistency between the raw image and the extracted front. The spatial resolution in our setup allows us to observe geometrical features for all scales down to pore scale, a. 2. Front analysis

The front is analyzed in terms of vertical extension, and statistical geometrical properties 共using the two-point correlation function and the box counting method兲. The descending coordinate along the maximum slope of the model is denoted z, the horizontal direction normal to z is called x. The frame (x,z) defines positions inside the model. Some of these positions, denoted ri ⫽(x i ,z i ), correspond to points that belong to the front. From the knowledge of these points, we define the following characteristic quantities for the front: its maximal extension along z, ⌬z, its width, w, defined as the root mean square of the z i distribution, and the two-point correlation function C(s), s being the scale of observation: C 共 s 兲 ⫽ 具具 ⌰ 共 r0 兲 ⌰ 共 r0 ⫹s兲典典兩 s兩 ⫽s ,

共1兲

where r0 is a point on the front and s is a vector from this point. ⌰(r) is 1 for points on the front and 0 otherwise. The double average denotes an average over all r0 on the front and all possible directions of s. For a fractal front with a fractal dimension D, it has been shown 关40兴 that C(s) is a power law of the scale s with an exponent 2⫺D. We also

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define the number N s of square boxes of side length s that are required for entirely covering the front. For a fractal front with a fractal dimension D, this box-counting method provides a power law with an exponent ⫺D 关41兴.

关25,27,29,46兴, but what happens in the crossover regime where gravity, capillary forces and viscous forces all are significant has with the exception of reservoir scale simulations 关7,39兴 not been studied. This crossover regime is what we are addressing in the following discussion.

III. COMPETITION BETWEEN GRAVITY AND VISCOUS FORCES

A. Study of a stable configuration

When one fluid invades another fluid with a different density, gravity forces tend to either stabilize or destabilize fluid displacement, depending on the density differences and flow directions 关25,27,28,33,42兴. We focus here on a situation where a less dense fluid with a negligible viscosity displaces a denser viscous fluid from above. In this case, the hydrostatic pressure gradient tends to annihilate any height difference between two points along the front, while local viscous effects tend to promote any part of the front that is progressing faster than the rest of it. The influence of gravity on the displacement structure depends both upon the gravity component on the model and upon the relative magnitudes of capillary and viscous forces. The latter relative magnitudes are usually quantified by use of the capillary number C a , which denominates the ratio of viscous forces to capillary forces at pore level:

␮va , ␥␬

1. Evolution of the pressure

We consider the situation where the front width reaches a limit value after a certain time, and only exhibits small fluctuations around this value afterward 共i.e., stable fronts兲. For constant flow rate the mean position of the front along the z direction, z f (z⫽0 at the outlet兲, would depend linearly on time, with a rate that is imposed by the withdrawal rate and the rate of cluster trapping. We assume that the viscous pressure gradient inside the model can be considered as the sum of an average uniform gradient and of local time-varying fluctuations, where the former term is given by the Darcy law: 兩 “ P 兩 ⫽ ␮ v / ␬ . As a result, the pressure at the outlet of the model fluctuates around a value given by the relation ¯P ⫽ P 0 ⫹ 共 ␳ g⫺ ␮ v / ␬ 兲 z f 共 t 兲 ,

共4兲

2

C a⫽

共2兲

where ␮ is the viscosity of the fluid, v is the filtering or Darcy velocity in the medium, a is the typical pore size, ␥ is the surface tension between the two fluids and ␬ is the permeability of the porous medium. Various values for the capillary number give rise to different displacement structures. The limit cases, i.e., capillary fingering for C a Ⰶ1 and viscous fingering for C a Ⰷ1, are well known. For displacements sufficiently slow for viscous forces to be negligible with respect to the fluctuations in threshold capillary pressures, the width of the invasion front is controlled by the latter fluctuations along the front. The dimensionless Bond number B o 关25–27兴, quantifies the competition between capillary forces and gravity at pore scale: ⌬ ␳ ga 2 , B o⫽ ␥

共3兲

where ⌬ ␳ is the difference in the two liquids’ densities and g is the component of gravity along the direction of displacement. The front width w for these slow displacements was ␯ /(1⫹ ␯ ) found to scale with the Bond number as w⬃B ⫺ 关25– o 27兴, where ␯ is the percolation correlation length exponent 关43兴, which for 2D systems is ␯ ⫽4/3. In another limit case in which a less viscous fluid invades a more viscous liquid at high capillary number, the displacement structure is controlled by viscous forces, which destabilize the invasion front. This viscous fingering regime has been studied both in terms of experiments 关14,18,44兴 and numerical simulations 关9,17,45兴. The effect of gravity has been studied for sufficiently low capillary numbers in both two and three dimensions

where P 0 is the pressure in the nonwetting fluid 共air兲, ␳ is the density of the defending fluid, ␮ is the viscosity of the defending fluid, v is the Darcy velocity, and ␬ is the model permeability. Therefore, under these assumptions the pressure P recorded at the model outlet as a function of time fluctuates around a linear trend, the slope of which is negative and proportional to ␳ g⫺ ␮ v / ␬ . Note that such a linear dependence of the outlet pressure is only expected for stable fronts. The conditions, in which these conditions are met, are explicited in Sec. III C. 2. Scaling of the front width

The following arguments follow the lines of 关25,26,30兴. Consider two different pores along the invasion front, labeled A and B and separated by a distance ␰ along the z direction 共see Fig. 4兲. If the density and viscosity of the invading nonwetting fluid 共air兲 are negligible, the pressure in the invading nonwetting fluid is constant and in the following we use this as our reference pressure P NW ⫽0. The capillary pressure over pore A is denoted P A . If we further assume that the viscous field in the defending wetting fluid along the front can be described by Darcy’s law, the capillary pressure over pore B, P B , can be written as the sum of hydrostatic and viscous pressure differences: v␮

␰. 共5兲 ␬ The pore throats in the porous medium have widths that are spatially uncorrelated. Each pore throat exhibits a capillary threshold pressure P t which is the minimum capillary pressure needed for the interface to penetrate the pore throat. For a throat with a meniscus at capillary pressure P c , the

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P B⫽ P A⫹ ␳ g ␰ ⫺



␳ ga⫺ F⫽

v␮a

␬

共10兲

.

Wt

The generalized fluctuation number F quantifies the ratio between the average pressure drop over pores and the fluctuations in capillary threshold pressures in the porous medium. It can be further related to the previously defined Bond and capillary numbers according to B o ⫺C a ⫽ FIG. 4. Sketch of a section of the front including two pores separated by a vertical distance ␰ . The capillary pressure over the pore labeled A is assumed to be the critical percolation threshold pressure P C . Assuming stable displacement, we can calculate the pressure at pore B.

criterion for invasion is P c ⬎ P t . The randomness of the porous medium gives a distribution N( P t ) for the capillary threshold pressure. Let us now consider an arbitrary throat in the medium and assume it has a meniscus with capillary pressure P c . The probability p that this throat will be invaded is p⫽p * ⫹

冕

Pc

P*

共6兲

N共 Pt兲d Pt ,

where p * is the occupation probability at the percolation capillary pressure P * for a flat model. P * would then be the lowest capillary pressure needed to produce a percolating cluster of invading fluid. Taylor-expanding N( P t ) around P * we get

For convenience we define a generalized bond number B * o as B* o ⫽B o ⫺C a .

A

v␮

␬

冊

␰ .

共8兲

Assuming that N( P t ) varies slowly around P * , we can estimate N( P * )⫽1/W t 关30兴, where W t is the width of the normalized capillary threshold pressure distribution N( P t ). With this assumption, Eq. 共8兲 can be rewritten as

␰ p⫺p * ⫽ F, a

␰ ⫽ 共 p⫺ p * 兲 ⫺ ␯ , a

共13兲

where ␯ is the correlation length exponent and A is a constant of order unity 关26,48兴. This scaling relation combined with Eq. 共9兲 yields Aw⫽A

冉冊

␰ ␰ ⫽ a a

w⬃F ⫺ ␯ /(1⫹ ␯ ) ⫽

共7兲

If we now let the capillary pressure over pore A in Eq. 共5兲 be the percolation threshold capillary pressure P * , we can use Eq. 共5兲 to rewrite Eq. 共7兲 as

冉

共12兲

The length ␰ is assumed now to be the correlation length of percolation clusters. Since the linear size of the largest trapped clusters of wetting fluid is limited by the front width w 关25,26,40,47兴, the correlation length is related to the front width in our problem as: w⫽ ␰ /a. From percolation theory we know 关43,47兴 that

and to the lowest order in ( P t ⫺ P * ) Eq. 共6兲 becomes

p⫺p * ⫽N 共 P * 兲 ␳ g ␰ ⫺

共11兲

⫺␯

F ⫺␯,

共14兲

which leads to the following scaling relation for the front width w:

N 共 P t 兲 ⫽N 共 P * 兲 ⫹N ⬘ 共 P * 兲共 P t ⫺ P * 兲 ⫹•••

p⫺ p * ⫽N 共 P * 兲共 P c ⫺ P * 兲 .

a W t F. ␥

共9兲

where the dimensionless generalized fluctuation number F 关30兴 has been introduced:

冉

␥ B* W ta o

冊

⫺ ␯ /(1⫹ ␯ )

.

共15兲

This relation is similar to the one obtained in capillary fingering under gravity 关25,30兴, but in our case the generalized fluctuation number F replaces the ordinary Bond number B o in the scaling relation 共15兲. It is important to note that F depends on the width of the capillary threshold distribution W t and that an increase in this width therefore gives an increase in the front width. In the experiments described in this paper, W t is kept constant. The range of validity of Eq. 共15兲 does not extend to situations where both viscous and gravitational forces exceed the capillary forces on pore scale. B. Limit of stability

The front width w in the scaling relation in Eq. 共15兲 diverges when the generalized Bond number goes to zero 共gravity forces balance viscous forces兲. This suggests a criterion for finite front widths and thus stable displacement when

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␳ g⫺

v␮

␬

⬎0.

共16兲

This is the same stability criterion as derived by Saffman and Taylor 关36兴 for the Hele-Shaw cell, though capillary forces in a Hele-Shaw configuration act at the scale of the system width L 共see Fig. 1兲, while in our porous medium they act at the pore scale a, which is much smaller than L. Section IV E addresses in more detail the differences between the mechanisms of the instability in our model and in a Hele-Shaw cell. C. Unstable displacement

Slow buoyancy driven unstable displacement has been studied in both two and three dimensions 关27,31,32,49兴. In this slow regime, string 共finger兲 like structures develop. Their characteristic width w f was found to scale with the Bond number in a similar fashion as the width of stable fronts: ␯ /(1⫹ ␯ ) . w f ⬃B ⫺ o

共17兲

From these results on buoyancy driven unstable systems we might conjecture that the above equation can be extended in the unstable case to include viscous effect, by replacing in Eq. 共17兲 B o with ⫺B o* : ⫺ ␯ /(1⫹ ␯ ) . w f ⬃ 共 ⫺B * o兲

共18兲

An experimental verification of this conjecture is the purpose of ongoing work. IV. RESULTS

With the setup described in Sec. II A, four main series of experiments were performed, corresponding to a total of 58 experiments: three series at constant capillary number C a ⯝82⫻10⫺3 , 130⫻10⫺3 , and 190⫻10⫺3 , and one series at constant Bond number B o ⯝0.155. Table I summarizes the experimental conditions for all experiments: inclination angle ␣ and withdrawal rate Q, and the corresponding capillary number C a , Bond number B o , and generalized Bond number B o* . The experiments are numbered by decreasing B* o . A. Observations of the displacement structure when varying B * o

Figure 5 shows the pictures of experiments for various capillary numbers and Bond numbers. The pictures shown here correspond to experiments labeled 4, 8, 13, 17, 22, 24, 25, 28, 29, 31, 46, 49, 50, 58 in Table I, from top to bottom and from left to right, respectively. Each column of the figure corresponds to the same capillary number C a , that is, to the same withdrawal speed applied during the experiment. The withdrawal rates in question are all large enough for the system to exhibit viscous fingering at zero inclination angle. Each line of the figure corresponds to the same generalized Bond number B * o ⫽B o ⫺C a . The maximum accessible Bond number is related to the maximum inclination angle possible with the setup 共around 55°). As a consequence, the maxi-

mum accessible B * o for a given capillary number decreases with C a , which explains why the highest values of B o* investigated here could not be reached for all values of C a . When increasing gravity for a given withdrawal speed 共thus, following a column in Fig. 5 from bottom to top兲, the displacement front is observed to evolve from narrow branched fingers to a nearly flat geometry, if the generalized Bond number B o* can be increased enough. Thus, viscous effects are responsible for a fingering of the displacement structure that can be stabilized by gravity. The similarity between all pictures on a given line of Fig. 5 suggests that the generalized Bond number is the controlling parameter for the process. This is consistent with relation 共15兲, which will be discussed further in Sec. IV D. Front destabilization at large withdrawal speed and low gravity appears more clearly in Fig. 6, which looks alike Fig. 5, except that each picture represents the whole dynamics of an experiment. Indeed, a gray level map displays the evolution of the displacement front with time. Time is the vertical descending coordinate, and each horizontal line on a map is a representation of the front at a given time. The coordinate along the t-line is the horizontal coordinate, x. The gray level at a given x along a given t-line corresponds to the difference between the position along z of the most advanced front point at horizontal position x and the mean front advancement along z. A constant gray level along a horizontal line means that the front is flat. Large variations of the gray levels along the line mean that some parts of the front 共corresponding to dark shadings兲 are left far behind others 共corresponding to light shadings兲. Thus, front destabilization appears as an increase of the roughness of the gray level map from the top to the bottom of the figure. Light shaded structures correspond to fingers development. These structures can widen as a result of the branching of the fingers, at low B * o . The same gray scale has been used throughout all pictures, which allows us to compare the importance of the fingering process in the various experiments. Pictures with homogeneous and small amplitude roughness correspond to experiments with a stabilized front. The two experiments with B * o ⬎0 fulfill this criterion. The intermittency of the dynamics for stable displacement is visible: a finger stops growing after a while, because its length has reached a value that causes gravity to stabilize it; then another finger grows, or the slow part of the front catches up fingers. As B o* is decreased below 0, gravity is not sufficient to prevent the fingers from growing forever, and, subsequently, to keep the front total vertical extent around a fixed value. This fingering becomes more important as B o* is further decreased. Thus, the crossover for the viscous instability in the presence of gravity seems to occur around B * o ⫽0, in good consistency with the theory exposed in Sec. III. B. Transition from stable to unstable drainage

The destabilization of the front for negative values of B o* clearly appears in Fig. 7共a兲, where the maximal extension of the front along the direction of displacement (z), ⌬z, is plotted as a function of time normalized by the time at breakthrough, for experiments at C a around 0.08. The maximal

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TABLE I. Summary of the inclination angles and withdrawal rates used for each of the 58 experiments. The corresponding characteristic numbers C a , B o , and B * o are also given. Depending on the withdrawal rate, the uncertainty on C a and, consequently, on B * o , not shown in this table, ranges between 0.009 and 0.025. Index

␣ 共deg兲 Q(ml/min) Bo Ca B* o

1

2

3

4

5

6

7

8

9

10

11

12

54.9 0.040 0.155 0.0032 0.1516

54.9 0.480 0.155 0.0389 0.1159

54.9 0.710 0.155 0.0581 0.0967

54.7 1.000 0.154 0.0826 0.0719

54.9 1.280 0.155 0.0988 0.0560

45.0 0.990 0.134 0.0787 0.0551

54.2 1.500 0.153 0.1193 0.0342

35.7 0.990 0.110 0.0825 0.0279

54.9 1.750 0.155 0.1351 0.0198

30.2 0.990 0.095 0.0779 0.0173

54.9 1.740 0.155 0.1424 0.0125

45.0 1.460 0.134 0.1341 ⫺0.0003

Index 13 14 15 16 17 18 19 20 21 22 23 24 ␣ 共deg兲 24.9 39.9 22.6 54.9 39.9 31.1 19.9 33.1 37.1 17.4 34.9 30.0 Q(ml/min) 0.980 1.480 1.000 2.260 1.540 1.480 1.030 1.480 1.530 1.000 1.510 1.450 Bo 0.080 0.121 0.073 0.155 0.121 0.098 0.064 0.103 0.114 0.057 0.108 0.095 Ca 0.0809 0.1245 0.0787 0.1622 0.1296 0.1142 0.0811 0.1211 0.1358 0.0803 0.1329 0.1231 ⫺0.0013 ⫺0.0031 ⫺0.0060 ⫺0.0074 ⫺0.0082 ⫺0.0165 ⫺0.0167 ⫺0.0177 ⫺0.0217 ⫺0.0237 ⫺0.0246 ⫺0.0285 B* o Index 25 26 27 28 29 30 31 32 33 34 35 36 ␣ 共deg兲 50.2 48.1 15.0 12.5 27.0 28.9 44.9 48.1 26.8 54.7 54.9 54.9 Q(ml/min) 2.020 2.000 1.040 1.000 1.480 1.500 2.020 2.060 1.610 2.070 2.520 2.270 0.145 0.141 0.049 0.041 0.086 0.091 0.134 0.141 0.085 0.154 0.155 0.155 Bo Ca 0.1778 0.1745 0.0859 0.0787 0.1257 0.1332 0.1778 0.1860 0.1330 0.2029 0.2042 0.2050 ⫺0.0324 ⫺0.0336 ⫺0.0369 ⫺0.0378 ⫺0.0397 ⫺0.0417 ⫺0.0442 ⫺0.0452 ⫺0.0476 ⫺0.0485 ⫺0.0494 ⫺0.0502 B* o Index 37 38 39 40 41 42 43 44 45 46 47 48 ␣ 共deg兲 44.9 20.3 10.0 44.9 44.9 43.7 5.0 14.9 10.0 0.0 35.6 28.1 Q(ml/min) 2.050 1.470 1.000 1.960 2.000 2.000 0.990 1.460 1.510 1.020 2.200 2.000 Bo 0.134 0.066 0.033 0.134 0.134 0.131 0.016 0.049 0.033 0.000 0.110 0.089 Ca 0.1867 0.1191 0.0880 0.1891 0.1961 0.1945 0.0841 0.1262 0.1235 0.0961 0.2072 0.1899 ⫺0.0531 ⫺0.0535 ⫺0.0551 ⫺0.0555 ⫺0.0625 ⫺0.0638 ⫺0.0676 ⫺0.0776 ⫺0.0907 ⫺0.0961 ⫺0.0970 ⫺0.1008 B* o Index 49 50 51 52 53 54 55 56 57 58 ␣ (deg) 5.2 21.1 15.0 0.0 0.0 54.7 15.0 9.3 32.3 0.1 Q(ml/min) 1.490 1.970 1.950 1.470 1.470 2.800 2.000 1.980 2.600 1.970 Bo 0.017 0.068 0.049 0.000 0.000 0.154 0.049 0.031 0.101 0.000 Ca 0.1208 0.1779 0.1626 0.1169 0.1169 0.2810 0.1992 0.1819 0.2750 0.1886 ⫺0.1036 ⫺0.1098 ⫺0.1136 ⫺0.1169 ⫺0.1169 ⫺0.1265 ⫺0.1502 ⫺0.1513 ⫺0.1739 ⫺0.1882 B* o

extension of the front is defined as the maximal vertical distance between two points on the front. At large positive generalized Bond numbers, the front extension rapidly grows to reach a constant plateau. Fluctuations of ⌬z occur around this average front extension due to the randomness of the medium. As the Bond number is decreased, these fluctuations grow. Viscous fingering features appear: ⌬z exhibits sudden drops that are characteristic of the merging of two nearby fingers. For negative values of B o* in the intermediate regime, the front extension has a growing trend with no saturation and important sudden drops due to finger merging. For large negative values of B * o 共not shown in Fig. 7兲, these drops disappear, as a small number of very thin fingers exist, that never merge. The crossover between stable displacement

and unbounded fingering appears to occur for a value of the generalized Bond number between ⫺0.006⫾0.009 and 0.017⫾0.013, in good consistency with the predicted value for the generalized Bond number at the crossover: B * o ⫽0. Also consistent with the predicted crossover is the evolution of the pressure drop across the model. Figure 7共b兲 shows the evolution of the pressure drop between the front 共atmospheric pressure兲 and the outlet channel of the model during the same experiments as in Fig. 7共a兲. For stabilized experiments (B * o ⬎0), it has been predicted to be linearly dependent on time. This can be seen in Fig. 7共b兲 for the four experiments with stable displacement. For B o* close to 0, the pressure drop shows little variations during drainage: gravity exactly balances the mean viscous pressure gradient. When

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FIG. 5. Pictures of the displacement structure for three series of experiments at capillary number C a ⯝0.081, 0.123, and 0.183, respectively. The provided ranges include the uncertainty on C a 共and subsequently, on B * o ) due to that on the viscosity measurements. The pictures represent the whole model (35⫻35 cm2 ). For each series, the displacement structure for an experiment with a generalized Bond number B o* around 0.072, 0.028, ⫺0.002, ⫺0.028, ⫺0.041, ⫺0.107, and ⫺0.189 共measured values兲 is shown. Stabilization of the front is observed for all series when sufficiently increasing B * o 共bottom to top兲; the crossover value of B * o for stabilization appears to be close to 0. The numbers inserted in the pictures refer to the index in Table I.

the average total pressure gradient turns positive, local effects in the viscous pressure gradient become significant, and nonlinearity appears in the pressure curves, while the trend of the pressure evolution changes from a decreasing to an increasing behavior.

FIG. 6. Evolution of the dynamics of the displacement when varying capillary number C a and generalized Bond number B o* . The values for these numbers are the same as in Fig. 5. Each picture displays the evolution of the displacement front with time. Axes for time and horizontal direction x are shown in the figure. The horizontal size of a picture corresponds to the model width 共35 cm兲. The vertical size corresponds to the time at breakthrough, which depends on the values for C a and B o . The gray level is proportional to the advancement of the front point in question with respect to the front mean position. Only the two top pictures, that exhibit a homogeneous roughness of small amplitude, correspond to experiments where the viscous instability is overcome by gravity. The numbers inserted in the pictures refer to the index in Table I. C. Geometrical properties of the front

The influence on drainage dynamics of the various physical effects involved appears when considering the evolution of the front fractal dimension during an experiment. Indeed, the limit cases in terms of displacement rate are well known. For very slow displacements, viscous effects have little influence on the interface. The displacement front for this capillary fingering regime is analogous to what is observed in inversion percolation. It is fractal, with a fractal dimension 1.33 关13兴. In contrast, viscous fingering occurs for very fast displacements, and leads to thin branched fingers that exhibit a fractal structure with a dimension that has been measured to 1.62⫾0.04 关14兴. The fractal dimensions measured in our experiments are consistent with these well-known values. Two families of fronts with behaviors close to gravity stabilized capillary fingering and viscous fingering, respectively, have been ana-

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FIG. 8. Characterization of two fronts corresponding respectively to capillary 共A兲 and viscous 共B兲 fingering, using the boxcounting method. The front labeled A corresponds to an experiment with a very slow withdrawal rate 共0.006 ml/min兲. Front B corresponds to an experiment with no gravity and a withdrawal rate of 5 ml/min. These two experiments are not referenced in Table I. N s is the number of boxes of side length s required for entirely covering the front. The observed trends for A and B are consistent with those expected for pure capillary 共slope ⫺1.33) and viscous fingering 共slope ⫺1.62), respectively.

FIG. 7. 共a兲 Evolution of the extension of the front along the direction of displacement (z), ⌬z, as a function of time normalized by the time at breakthrough. 共b兲 Evolution of the pressure drop between the front and the model outlet channel, as a function of normalized time also. The uncertainties on B o* are shown only in 共b兲. Both plots are consistent with the theoretical prediction of a crossover at B * o ⫽0.

lyzed using both the two-point correlation function and the box-counting algorithm, with consistency. The results of the box-counting method are shown in Fig. 8. The box-counting function for family A 共capillary fingering兲, shown in Fig. 8, has been obtained by averaging over all stabilized fronts in the corresponding experiment. The box-counting function for family B 共viscous fingering兲 is the box-counting function for a sole front because of the unstable growth. At scales smaller than a crossover scale s⬍s c ⯝20 pore sizes, fronts A exhibit a regime consistent with that observed for invasion percolation, with a fractal dimension close to 1.3. The crossover scale, s c , is consistent with the observed front width. At scales larger than s c , the front is seen as line. Front B displays a fractal structure with a fractal dimension close to 1.6, for scales larger than 2 pore scales and smaller than 80 pore scales. Front B also exhibits some broadening of the fingers at small scales, related to capillary effects. By ‘‘small

scales,’’ we mean scales smaller than the size of the clusters trapped inside the branched fingers. For the intermediate regimes in which viscous forces and gravity are of the same order of magnitude, parts of the front display fingers controlled by viscous effects. The finger tips progress faster than other parts of the front, which are controlled by the balance between capillary forces and gravity 共see Fig. 3兲. The evolution of the correlation function on the whole range of scales during an experiment gives hints on the dynamics of the displacement process. In Fig. 9共b兲, density-density correlation functions for a series of experiments in stable regime and corresponding to various values of the generalized Bond number, B * o , have been plotted as a function of the scale normalized by the mean front width, w. One front from each of these experiments is shown in Fig. 9共a兲. They come from seven different experiments in the stable regime with B o ⫽0.154 but with different injection rates. By increasing the injection rate, the pressure gradients in the fluid decrease, which yields a lower B * o and therefore a wider front. At the largest speeds the fronts look visually somewhat different from the fronts observed for pure capillary stabilized fingering. The depth of the fjords are typically larger than their width. The correlation functions in Fig. 9共b兲 were obtained by averaging over all stable fronts from the experiments in question. The correlation function is divided by s ⫺0.67, which is the expected scaling in the capillary fingering regime. The correlation function corresponding to the slowest withdrawal rate 共experiment 1兲 exhibits at small scales a clear horizontal line characteristic for capillary fingering. The horizontal line C is a guide to the eye. At higher injec-

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FIG. 10. Scaling of the front width as a function of the generalized Bond number B * o for a series of experiments in stable configuration at B o ⯝0.154. The regression is consistent with a power law with an exponent ⫺0.55, in good agreement with the theoretically predicted relation 共15兲. Vertical error bars represent the uncertainty on the determination of the mean front width for a given experiment. Horizontal error bars are related to the uncertainty in the viscosity measurements.

scales larger than the initial capillary regime 共the part on small length scales with a slope close to zero兲, but smaller than the front width. It is important to note that all these fronts are in the stable regime. As a guide to the eye we have also plotted the slope 0.3 共see long-dashed line V), which is the expected slope for unstable viscous fingers. As seen in the figure, the effective slope observed in the viscous crossover regime is larger than the slope characteristic of pure capillary fingering, but smaller than that characteristic of pure viscous fingering. At scales larger than the width of the fronts, the fronts become effectively linear with a slope corresponding the dotted line L, which is the expected result for a linear front. D. Scaling of the front width for stable displacements FIG. 9. 共a兲 Fronts taken from experiments at B o ⯝0.154 and for 6 different displacement rates, all corresponding to positive generalized Bond numbers B * o 共stable displacement兲. 共b兲 Corresponding correlation functions for these experiments. Each function has been obtained by averaging over all stable fronts. The scale is normalized by the measured front width, w, which, according to relation 共15兲, is ⫺4/7 expected to scale as B * 共see Fig. 10 for a confirmation of this o prediction兲. The horizontal line C corresponds to the capillary fingering behavior, the long-dashed line V to the pure viscous fingering behavior, and the dotted line L to the linear shape of the front. Experiment 1 is close to pure capillary fingering 关as exposed in Fig. 8 共case A兲兴. Due to viscous effects, the dynamics of the displacement is not that of invasion percolation for regimes close to the instability threshold.

tion rates the correlation function displays a crossover which becomes more significant as the injection rate is increased. This crossover is the Hallmark of significant viscous effects and is seen as an increase in the effective slope on length

In the case of stable displacement, Eq. 共15兲 predicts a scaling of the front width as a power law of the generalized Bond number, B o* ⫽B o ⫺C a , with an exponent ⫺ ␯ /(1⫹ ␯ ) ⫽⫺4/7 (⬇⫺0.57). Figure 10 shows the evolution of the front width, w, as a function of B * o , for a series of experiments at Bond number B o ⫽0.154⫾0.01. The experiments in question are indexed 1, 2, 3, 4, 5, 7, and 9 in Table I. The values for the front width have been obtained by averaging over all stabilized fronts from the corresponding experiment. The error bars account for the sensitivity of the time averaging on the choice of the time after which the fronts are supposed to be stabilized. The dependence of w on B * o is consistent with a power law with an exponent ⫺0.55, in good agreement with the theoretical predictions. E. Discussion

The viscous instability observed in the present article appears for a magnitude of gravity identical to that of the vis-

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cous forces. Such a crossover was also observed by Saffman and Taylor in a Hele-Shaw cell with no porous medium under the same conditions. However, though the forces responsible for the stability or instability of the interface are of identical nature, the resulting displacement processes are significantly different in the two geometries. The first striking difference lies in the nature of stable displacement in the two configurations. A stable displacement in the Hele-Shaw cell configuration means that the front exhibits no roughness. Observed in the uniformly moving referential attached to its mean position, the front is simply static. In our porous model, in contrast, an initial front roughness appears due to capillary fluctuations; during later stages of the drainage process, this roughness develops due to both local viscous effects and inhomogeneous capillary forces. The resulting displacement front has a significant roughness in the direction of displacement. Observed in the moving frame attached to its mean position, the front is ever changing, but with an amplitude that saturates after a certain time to a value that can be as high as 50 pores scale units. Thus, stable displacement consists here in an ongoing competition process where departing viscous fingers are prevented by gravity from developing too much ahead of the front mean position on behalf of other fingers or slowly moving portions of the front. The front width is controlled by the ratio of the mean effective pressure gradient inside the model 共i.e., hydrostatic pressure gradient minus viscous pressure gradient兲 to the inhomogeneous capillary forces. The mean effective pressure gradient is denoted by the generalized Bond number B * o ⫽B o ⫺C a . The inhomogeneous capillary threshold pressures result from the randomness in the porous media. This inhomogeneity could be quantified by the width W t of the capillary threshold pressure distribution N( P t ). In our case, it is measured to be W t ⬇ ␥ /(4a). The ratio between the effective pressure gradient and the fluctuations in capillary threshold pressure distribution is quantified by the dimensionless fluctuation number F defined in Eq. 共10兲. Due to the homogeneity of our porous model, W t happens to be constant in space. For a medium with a varying pore scale, Eq. 共15兲 indicates possible significant changes in the front width during the displacement process for constant flow rate and gravity component on the model. In the framework of this study, the extensively studied capillary and viscous fingering appear as limit cases where the capillary number, C a , or Bond number, B o , respectively, are negligible. The scaling relation obtained for the front width in stable displacement as a function of F, and thus, of B o* , is therefore a generalization of that previously obtained for capillary fingering. Indeed, Bond number and generalized Bond number are identical when the capillary number is negligible. Second, the transition from stable to unstable displacement in our system does not consist of a radical change in the local dynamics of the interface, as for viscous instability observed in a Hele-Shaw cell. For a configuration close to the instability threshold, it is difficult to know, by looking at the local dynamics of the displacement in our experimental model, whether the front amplitude is going to reach a saturation value or whether it is going to grow forever. This is also visible from the large fluctuations in the front width

evolution for configurations close to the instability threshold, in Fig. 7: do these large fluctuations diverge or do they oscillate around a constant value? In the case of the Hele-Shaw cell configuration, in contrast, the instability can be inferred as soon as a significant roughness appears in the front. In a sense, the complexity of the porous medium translates into a complexity in the viscous instability. V. CONCLUSION

We have studied experimentally the displacement of a mixture of glycerol and water by air at constant volumetric flow rate in a synthetic random two-dimensional medium. The experimental setup allowed the tuning of gravity by tilting the setup and of viscous effects by changing the withdrawal rate. The aim of the study was to investigate the crossover regime from capillary fingering to viscous fingering during drainage with gravity. To account for the relative importance of viscous- and gravitational effects during drainage, we have introduced the fluctuation number F which is defined as the ratio of the effective fluid pressure drop 共i.e., average hydrostatic pressure drop minus viscous pressure drop兲 at pore scale to the width of the fluctuations in the threshold capillary pressures. We observe a crossover at F ⫽0 between a configuration where the displacement front keeps a finite extension along the direction of flow (F⬎0) and a situation where fingers grow ahead of other parts of the front, forever (F⬍0). Gravity stabilized capillary fingering and pure viscous fingering appear as limit cases where viscous effects or gravity effects, respectively, have little influence on the displacement structure. In intermediate regimes (F close to 0兲, the dynamics of the displacement seems to hold features characteristic of invasion percolation 共for short length scales of the front兲 as well as features characteristic of viscosity-controlled fingering. The crossover is smooth and might lead to an apparent misleading dimension of the front in a midrange between capillary regime and viscous fingering dimension. The drainage process is both inhomogeneous in time and space. For positive values of the fluctuation number, the front width has a physical meaning in terms of fluctuations of the front around a mean position that progresses at constant speed. This front width is controlled by viscous effects and scales as a power law of the fluctuation number with an exponent ⫺4/7 关see Eq. 共15兲兴, a scaling law that we predict theoretically and that is nicely confirmed by the experiments. For negative values of the fluctuation number, characteristic length scales are difficult to find in the problem. The width of the developing fingers might be a suitable characteristic scale, with a scaling relation identical to Eq. 共15兲 as a function of the generalized Bond number’s 共or, equivalently, the fluctuation number’s兲 absolute value. ACKNOWLEDGMENTS

The work was supported by NFR, the Norwegian Research Council, VISTA, the Norwegian Academy of Science and Letters’ research program with Statoil and the French/ Norwegian collaboration PICS.

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